REAL NUMBERS

CHAPTER 2:      REAL NUMBERS

2.1 THE SET OF REAL   NUMBERS   SYSTEM

                                          The Real Number System

Each real number is a member of one or more of the following sets.
The sets of numbers described in the following table should look familiar to you.  It is sometimes handy to have names for these sets of numbers, so knowing their names can simplify, for example, describing domains of functions or comprehending theorems such as the rational zeros theorem.

Set	Description
Natural numbers	{1, 2, 3, 4, …. }
Whole numbers	{0, 1, 2, 3, 4, …}
Integers	{ …, -3, -2, -1, 0, 1, 2, 3, …. }
Rational numbers	All numbers that can be written as, where a and b are both integers, and b is not equal to 0.
Irrational numbers	Numbers such as
Real numbers	The union of the sets of rational numbers and irrational numbers

Things to notice:

• The set of Whole numbers is the same as the set of Natural numbers, except that it includes 0.  To help remember this, think “o” is in “whole.”

• The set of Integers is the same as the set of whole numbers and the negatives of the whole numbers.

• We can think of Rational numbers as fractions.  To remind us, notice that the word “ratio” is embedded in the word “rational.”  A ratio is a fraction.

• The set of Rational numbers includes all decimals that have either a finite number of decimal places or that repeat in the same pattern of digits.  For example, 0.333333… = 1/3 and .245245245…. = 245/999.

• The set of Natural numbers is a subset of the set of Whole numbers, which is contained in the set of Integers, which is inside of the set of Rational numbers.

Example

Classify the following numbers.  Remember that a number may belong to more than one category.
            0,   4,   -9,  0.23, 

Solution 

Number                     Member of these sets

0                                  Whole, Integer, Rational (can be written as ), Real

4                                  Natural, Whole, Integer, Rational (can be written as ), Real

-9                                 Integer, Rational, Real

                                Rational, Real

                              Natural (), Whole, Integer, Rational, Real

                            Integer ( ), Rational, Real

                            Irrational  (≈ 3.31662479036…  This is not a terminating
                                    decimal and it does not repeat), Real

                                Whole (), Integer, Rational, Real

                       Rational ( =  ) , Real

                                Irrational, Real

                             Rational, Real

0.23                            Rational (terminating decimal equal to ), Real

                             Irrational (It is a fraction, but not a quotient of two integers), Real

2.2 INTEGERS MODELS

                      Modeling integers

When modeling integers, we can use colored chips to represent integers. One color can represent a positive number and another color can represent a negative number 

Here, a yellow chip will represent a positive integer and a red chip will represent a negative integer

For example, the modeling for 4, -1, and -3 are shown below:

It is extremely important to know how to model a zero. Basically, if we have the same amount of yellow chips and red chips we say that we have zero pair(s) 

For example, all the followings represent zero pair(s)

And so on...


Adding and subtracting integers with modeling can be extremely helpful if you are having problems understanding integers

In modeling integers, adding and subtracting are always physical actions.

If a board is used with the chip, adding always mean " Add something to the board" and subtraction always mean "Remove something from the board"

Here, we will use a big square to represent a board

Let's start with addition of integers:

Example #1: -2 + -1

Put two red chips on the board. Then put one red chip on the board. Since we end up with 3 red chips, the answer is -3

Notice that big arrow represents the "+" sign or the action of adding

Example #2: -3 + 2

Add 3 red chips on the board to represent -3. Then, add 2 yellow chips to represent 2

Remove the two zero pairs from the board. Since only one red chip remains, the answer is -1

Example #3: -4 − - 2

Put 4 red chips on the board to represent -4. Then, the problem says that you have to minus negative two

Minus negative two means as we said before to remove -2 from the board. In other words, it means to remove two red chips from the board.Thus, the answer is -2

Subtracting with chips becomes tricky when what they tell you to subtract does not exist

Example #4: -4 − 2

Here, the problem is not asking to subtract -2 as before, but 2

Start by putting 4 red chips to represent -4

Now how do you remove 2 or two yellow chips that you don't have?

The only way to do it is to add two zero pairs to the board ( Shown in the board on the right)

Now,you can remove the two yellow chips. After you do that, you are left with 6 red chip, so the answer is -6

I tried to model most situations above. However, I recognize the fact that modeling integers can be a bit tricky if you have never seen this before.
                                 

2.3 COMPARING AND ORDERING INTEGERS

      Learn to compare and order integers..

   Remember!    
                       Numbers on a number line increase in value as you move from left to right       

Additional Example 1: Comparing Integers                                                             
  Use the number line to compare each pair of integers. Write < or >.

                  
                

      A. –2         2

              –2 < 2      –2 is to the left of 2 on the number line

     B.  3      –5

               3 > –5      3 is to the right of –5 on the number line

   C. –1       –4

              –1 > –4     –1 is to the right of –4 on the number line

Additional Example 2: Ordering Integers   

Order the integers in each set from least to greatest.

A. –2, 3, –1

    Graph the integers on the same number line.

                    

      Then read the numbers from left to right: –2, –1, 3.

B. 4, –3, –5, 2

       Graph the integers on the same number line.

 

Then read the numbers from left to right: –5, –3, 2, 4.

2.4 ABSLUTE VALUES

Absolute Value

Absolute Value means ...

... only how far a number is from zero:

"6" is 6 away from zero,  and "-6" is also 6 away from zero. So the absolute value of 6 is 6,  and the absolute value of -6 is also 6

More Examples:

• The absolute value of -9 is 9
• The absolute value of 3 is -3
• The absolute value of 0 is 0
• The absolute value of -156 is 156

No Negatives!

So in practice "absolute value" means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero).

Absolute Value Symbol

To show that you want the absolute value of something, you put "|" marks either side (they are called "bars" and are found on the right side of your keyboard), like these examples:

|-5| = 5

|7| = 7

Sometimes absolute value is also written as "abs()", so abs(-1) = 1 is the same as |-1| = 1

Subtract Either Way Around

And it doesn't matter which way around you do a subtraction, the absolute value will always be the same:

|8-3| = 5	|3-8| = 5
(8-3 = 5)	(3-8 = -5, and |-5| = 5)

 

2.5 PROPERTIES OF REAL NUMBERS

   Basic Algebraic Properties:
Let  and denotes real numbers.

(1) The Commutative Properties
(a)  (b) 
The commutative properties says that the order in which we either add or multiplication real number doesn’t matter.

(2) The Associative Properties
(a)  (b) 
The associative properties tells us that the way real numbers are grouped when they are either added or multiplied doesn’t matter. Because of the associative properties, expressions such as  and makes sense without parentheses.

(3) The Distributive Properties
(a)  (b) 
The distributive properties can be used to expand a product into a sum, such as or the other way around, to rewrite a sum as product:

(4) The Identity Properties
(a)  (b) 
We call  the additive identity and  the multiplicative identity for the real numbers.

(5) The Inverse Properties
(a) For each real number , there is real number , called the additive inverse of , such that 
(b) For each real number , there is a real number , called the multiplicative inverse of , such that
Although the additive inverse of , namely , is usually called the negative of , you must be careful because  isn’t necessarily a negative number. For instance, if , then . Notice that the multiplicative inverse  is assumed to exist if . The real number  is also called the reciprocal of  and is often written as .

Example:
State one basic algebraic property of the real numbers to justify each statement:
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) If , then 

Solution:
(a) Commutative Property for addition
(b) Associative Property for addition
(c) Commutative Property for multiplication
(d) Distributive Property
(e) Additive Inverse Property
(f) Multiplicative Identity Property
(g) Multiplicative Inverse Property

Many of the important properties of the real numbers can be derived as results of the basic properties, although we shall not do so here. Among the more important derived propertiesare the following.

(6) The Cancellation Properties:
(a) If  then, 
(b) If  and , then 

(7) The Zero-Factor Properties:
(a) 
(b) If , then  or  (or both)

(8) Properties of Negation:
(a) 
(b) 
(c) 
(d) 

Subtraction and Division:
Let  and  be real numbers,
(a) The difference  is defined by 
(b) The quotient or ratio  or  is defined only if . If , then by definition 
It may be noted that Division by zero is not allowed.
When  is written in the form , it is called a fraction with numerator  and denominator . Although the denominator can’t be zero, there’s nothing wrong with having a zero in the numerator. In fact, if , 

(9) The Negative of a Fraction:
If , then 

2.6 -2.9    OPERATIONS IN INTEGERS (ADDITION,SUBTRACTION,MULTIPLICATION, & DIVIDING INTEGERS)

	Operations on Integers
	Addition
	There are three appealing ways to understand how to add integers. We can usemovement, temperature and money. Lastly, we will take a look at the rules for addition. Movement You are probably familiar with a number line (see below). Traditionally, zero is placed in the center. Positive numbers extend to the right of zero and negative numbers extend to the left of zero. In order to add positive and negative integers, we will imagine that we are moving along a number line. Number Line ex 1: If asked to add 4 and 3, we would start by moving to the number 4 on the number line -- exactly four units to the right of zero. Then we would move three units to the right. Since we landed up seven units to the right of zero as a result of these movements, the answer must be 7. ex 2: If asked to add 8 and -2, we would start by moving eight units to the right of zero. Then we would move two units left from there because negative numbers make us move to the left side of the number line. Since our last position is six units to the right of zero, the answer is 6. ex 3: If asked to add -13 and 4, we start by moving thirteen units to the left of zero. Then we move four units to the right. Since we land up nine units to the left of zero, the answer is -9. ex 4: If asked to add -6 and -5, first move six units to the left of zero. Then move five units further left. Since we are a total of eleven units left of zero, the answer is -11. Temperature The temperature model for adding integers is exactly the same as the movement model because most thermometers are really number lines that stand upright. The numbers can be thought of as temperature changes. Positive numbers make the temperature indicator rise. Negative numbers make the temperature indicator fall. Adding two positive temperatures will result in a positive temperature, similar to example 1 above. Adding two negative temperatures will result in a negative temperature, similar to example 4 above. Examples 2 and 3 can be understood in a different way by imagining a battle between two temperatures. When we added 8 and -2 in example 2, there was more positive temperature than negative temperature which would explain the result -- positive 6. In example 3 there was more negative temperature than positive. That will explain why the answer is negative. Money It can be helpful to think of money when doing integer addition. The positive numbers represent income while the negative numbers represent debt. When adding two incomes, like example 1 above, the answer has to be a bigger income and the result is a positive number. When adding two debts, like example 4 above, the answer has to be another debt. In fact, accountants would call it 'falling deeper in debt.' Similar to our temperature battle between warm temperatures and cold temperatures, adding positive and negative numbers is like comparing income to debt. If there is more income than debt the answer will be positive, like example 2. If there is more debt than income the answer will be negative, like example 3. Rules for Addition Below is a table to help condense the rules for addition. Note the second and third rows of the body of the table. Those answers are dependant upon the original values. Rules for Addition Positive + Positive Positive Positive + Negative Depends Negative + Positive Depends Negative + Negative Negative Quizmaster: Adding Integers
	Subtraction
	Instead of coming up with a new method for explaining how to subtract integers, let us borrow from the explanation above under the addition of integers. We will learn how to transform subtraction problems into addition problems. The technique for changing subtraction problems into addition problems is extremely mechanical. There are two steps: 1.        Change the subtraction sign into an addition sign. 2.        Take the opposite of the number that immediately follows the newly placed addition sign. Let's take a look at the problem 3 - 4. According to step #1, we have to change the subtraction sign to an addition sign. According to step #2, we have to take the opposite of 4, which is -4. Therefore the problem becomes 3 + (-4). Using the rules for addition, the answer is -1. Here is another problem: -2 - 8. Switching the problem to an addition problem, it becomes -2 + (-8), which is equal to -10. 6 - (-20) is equal to 6 + 20, which is 26. -7 - (-1) is the same as -7 + 1, which is -6. Quizmaster: Subtracting Integers
	Multiplication
	The best place to start with multiplication, is with the rules: Rules for Multiplication Positive x Positive Positive Positive x Negative Negative Negative x Positive Negative Negative x Negative Positive Now we have to understand the rules. The first rule is the easiest to remember because we learned it so long ago. Working with positive numbers under multiplication always yeilds positive answers. However, the last three rules are a bit more challenging to understand. The second and third steps can be explained simultaneously. This is because numbers can be multiplied in any order. -3 x 7 has the same answer as 7 x -3, which is always true for all integers. [This property has a special name in mathematics. It is called the commutative property.] For us, this means the second and third rules are equivalent. One reason why mathematics has so much value is because its usefulness is derived from its consistency. It behaves with strict regularity. This is no accident, mind you. This is quite purposeful. Keeping this in mind, let's take a look at Figure 2 below. There is a definite pattern to the problems in the table. The first number in each row remains constant but the second number is decreasing by one, each step down the table. Consequently, the answer is changing. The answers have a definite pattern as we go down the table too. It should be relatively easy to determine the two missing answers. If you understand the pattern, you will see that the first unanswered problem is -2 and the second unanswered problem is -4. This should provide some meaning why a negative number is always the result when multiplying two numbers of opposite sign. Likewise, lets turn our attention to Figure 3 below. This table has a pattern similar to the one in Figure 1. However, this table begins with a negative number. As we scan the list of answers, we can see that the last two problems remain unanswered. With a little concentration, we can see that the two unanswered questions must have positive answers to maintain mathematical consistency. This should help us understand why a positive number is always the result of multiplying two numbers of the same sign.  Figure 2 2 x 3 = 6 2 x 2 = 4 2 x 1 = 2 2 x 0 = 0 2 x -1 = ? 2 x -2 = ? Figure 3 -8 x 3 = -24 -8 x 2 = -16 -8 x 1 = -8 -8 x 0 = 0 -8 x -1 = ? -8 x -2 = ? Here are some examples: 4 x -8 = -32, -6 x 8 = -48, and -20 x -3 = 60. Quizmaster: Multiplying Integers
	Division
The rules for division are exactly the same as those for multiplication. If we were to take the rules for multiplication and change the multiplication signs to division signs, we would have an accurate set of rules for division. For example, -9 ÷ 3 = -3. 20 ÷ (-4) is equal to -5. -18 ÷ (-3) is equal to 6 2.10   POWERS OF INTEGERS The power of a natural exponent of an integer is another integer. The absolute value of the result is the absolute value of the base multiplied by itself as specified by the exponent. The sign of the result can be determined by the following rule: 1. The powers of an even exponent are always positive. 2. The powers of an odd exponent have the same sign of the base. Properties 1. a0 = 1 2. a1 = a 3. Multiplication of powers with the same base: It is another power with the same base and the exponent is the sum of theexponents. am · a n = am+n (−2)5 · (−2)2 = (−2)5+2 = (−2)7 = −128 4. Division of powers with the same base: It is another power with the same base and the exponent is the difference of the exponents. am : a n = am — n (−2)5 : (−2)2 = (−2)5 — 2 = (−2)3 = −8 5. Power of a power: It is another power with the same base and the exponent is the product of the exponents. (am)n = am · n [(−2)3]2 = (−2)6 = 64 6. Multiplication of powers with the same exponent: It is another power with the same exponent, whose base is the product of the bases. an · b n = (a · b) n (−2)3 · (3)3 = (−6)3 = −216 7. Division of powers with the same exponent: It is another power with the same exponent, whose base is the quotient of the bases. an : b n = (a : b) n (−6)3 : 33 = (−2)3 = −8 Powers with an Integer Exponent Powers with a Fractional Exponent Properties 1, a0 = 1 2 a1 = a 3. Multiplication of powers with the same base: It is another power with the same base and whose exponent is the sum of the exponents. am · a n = am+n (−2)5 · (−2)2 = (−2)5+2 = (−2)7 = −128 4. Division of powers with the same base: It is another power with the same base and the exponent is the difference between the exponents. am : a n = am — n (−2)5 : (−2)2 = (−2)5 — 2 = (−2)3 = −8 5. Power of a power: It is another power with the same base and the exponent is the product of the exponents. (am)n = am · n [(−2)3]2 = (−2)6 = 64 6. Multiplication of powers with the same exponent: It is another power with the same exponent, whose base is the product of the bases an · b n = (a · b) n (−2)3 · (3)3 = (−6)3 = −216 7. Division of powers with the same exponent : It is another power with the same exponent, whose base is the quotient of the bases. an : b n = (a : b) n (−6)3 : 33 = (−2)3 = −8 Negative Exponents 2.11 RATIONAL NUMBERS          Rational Numbers A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. The number 8 is a rational number because it can be written as the fraction 8/1. Likewise, 3/4 is a rational number because it can be written as a fraction. Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction. 2.12 DENSITY PROPETY OF REAL NUMBERS Density property The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, 5.613 and so forth. Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers! Between any two real numbers, there is always another real number 2.13  RATIONAL NUMBERS OF DECIMALS 2.14  EXPRESSING A FRACTION AS A QUOTIENT OF TWO INTEGERS 2.15  SQUARE ROOTS Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers. The square root of a number, n, written below is the number that gives n when multiplied by itself. Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 … The square root of a number, n, written  is the number that gives n when multiplied by itself. For example, because 10 x 10 = 100 Examples Here are the square roots of all the perfect squares from 1 to 100. Finding square roots of of numbers that aren't perfect squares without a calculator 1. Estimate - first, get as close as you can by finding two perfect square roots your number is between. 2. Divide - divide your number by one of those square roots. 3. Average - take the average of the result of step 2 and the root. 4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you. Example: Calculate the square root of 10 () to 2 decimal places. 1. Find the two perfect square numbers it lies between. Solution: 32 = 9 and 42 = 16, so  lies between 3 and 4. 2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer) 3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667 Repeat step 2: 10/3.1667 = 3.1579 Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623 Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001 If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3. Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.   Example: Calculate the square root of 10 () to 2 decimal places. 1. Find the two perfect square numbers it lies between. Solution: 32 = 9 and 42 = 16, so  lies between 3 and 4. 2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer) 3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667 Repeat step 2: 10/3.1667 = 3.1579 Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623 Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001 If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3. 2.16 PROPERTIES OF SQUARE ROOTS  Square Root Property The square root property is one method that is used to find the solutions to a quadratic (second degree) equation.  This method involves taking the square roots of both sides of the equation.  Before taking the square root of each side, you must isolate the term that contains the squared variable.  Once this squared-variable term is fully isolated, you will take the square root of both sides and solve for the variable.  We now introduce the possibility of two roots for every square root, one positive and one negative.  Place a  sign in front of the side containing the constant before you take the square root of that side. Example 1:      … the squared-variable term is isolated, so we will take the square root of each side                     … notice the use of the  sign, this will give us both a positive and                                                a negative root          … simplify both sides of the equation, here x is isolated so we have                                                solved this equation  Example 2:                   … again the squared-variable term is isolated, so we will take the                                              square root of each side          … again don’t forget the  sign, now simplify the radicals                       … this time p is not fully isolated, also notice that 4 are rational                                                          numbers, which means …  and   and  Example 3:               … squared term is not isolated, add 1 to each side before beginning         … now take the square root of both sides               … simplify radicals        … radical containing the constant cannot be simplified, solve for the variable        … notice the placement of the –1 before the radical on the right-hand                                                          side, these numbers may not be combined since –1 is a rational                                                          number and  are irrational numbers   In each of the (above) 3 examples involving the square root property, notice that there were no first-degree terms.  These equations although they are quadratic in nature, have the form or Square Root Rules Algebra rules for square roots are listed below. Square root rules are a subset of nth root rules andexponent rules. Definitions 1.  if both b ≥ 0 and b2 = a. 2.  Examples  because 32 = 9. 3. If a ≥ 0 then . Distributing (a ≥ 0 and b ≥ 0) 1.  2.      (b ≠ 0) 3.  Examples 4.  Rationalizing the Denominator (a > 0, b > 0, c > 0) Examples Careful!! 1.  2.  3.  Examples 2.17 IRRATIONAL NUMBERS Irrational Numbers An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means not Rational Examples: Rational Numbers OK. A Rational Number can be written as a Ratio of two integers (ie a simple fraction). Example: 1.5 is rational, because it can be written as the ratio 3/2 Example: 7 is rational, because it can be written as the ratio 7/1 Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3 Irrational Numbers But some numbers cannot be written as a ratio of two integers ... ...they are called Irrational Numbers. It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy! Example: π (Pi) is a famous irrational number. π = 3.1415926535897932384626433832795 (and more...) You cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571... is close but not accurate. Another clue is that the decimal goes on forever without repeating. Rational vs Irrational So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction. Example: 9.5 can be written as a simple fraction like this: 9.5 = 19/2 So it is a rational number (and so is not irrational) Here are some more examples: Number As a Fraction Rational or Irrational? 1.75 7/4 Rational .001 1/1000 Rational √2  (square root of 2) ? Irrational ! Square Root of 2 Let's look at the square root of 2 more closely. If you draw a square of size "1",  what is the distance across the diagonal? The answer is the square root of 2, which is 1.4142135623730950...(etc) But it is not a number like 3, or five-thirds, or anything like that ... ... in fact you cannot write the square root of 2 using a ratio of two numbers ... I explain why on the Is It Irrational? page, ... and so we know it is an irrational number Famous Irrational Numbers Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...) The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...) Many square roots, cube roots, etc are also irrational numbers. Examples: √3 1.7320508075688772935274463415059 (etc) √99 9.9498743710661995473447982100121 (etc) But √4 = 2 (rational), and √9 = 3 (rational) ... ... so not all roots are irrational.   Note on Multiplying Irrational Numbers Have a look at this: π × π = π2 is irrational But √2 × √2 = 2 is rational So be careful ... multiplying irrational numbers might result in a rational number! History of Irrational Numbers Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational. However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!